2 Answers
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They are not exactly interchangeable, but the distinction is very subtle. To illustrate, let me first change your example sentences into the forms I find most agreeable.

The people were free of the barbaric dictator.

The mashed potatoes were free of lumps.

I wish I could be rid of this terrible cold.

This demonstrates that "free of" is most comfortable for me when used to indicate that something no longer is beset by an entity that had been pervasively enmeshed in its very existence, as a dictator controls every facet of a people's lives, as the lumps in mashed potatoes influence every bite of the food.

"Free from," on the other hand, seems to indicate more of an escape in the case of something that was more externally attached, as in, "Now at last I am free from the annoying attention of my mother, who wouldn't leave me alone until I moved to New York."

As for the last example, "rid of this terrible cold," I find "rid" to work better with an invader, such as a disease, or anything that is more "invasive" as opposed to "pervasive." In other words, I would say: "I need to get rid of that tick that's been biting me all day." I would not say: "I need to be free from that tick." And that's why I would use "rid of" with a cold, rather than "free" (no matter whether of or from).

Let me take it a bit further. If we extend the conceptualization to the word "freedom," I think we'll find more basis for differentiation in the choices between "free of" and "free from." So let's try a few examples.

"Freedom from want." "Freedom from fear." "Freedom from hunger." These phrases cannot be constructed using the word "of." They demonstrate of being free from an entity that is externally attached in a conceptually philosophical way; hunger besets you, fear comes upon you, "want" sinks its claws into you. If you can remove these things from your life, you are "free from" the undesirable attention (attack) of these things.

And one more distinction that I find to be quite telling:

"Free of" is best used with nouns ("At last I'm free of those thoughts), whereas "free from" works much better with verb forms ("At last I'm free from thinking that way). It just doesn't work well for me the other way around.

I don't have a definitive answer to this question. But I want to point out a couple of things that surprised me when I looked into possible differences between "free of" and "free from."

First, I ran an Ngram Viewer search across the years 1800 to 2007, of four phrases: "are free of," "are free from," "is free of," and "is free from." Since these are all fairly common phrases today, I wanted to see whether they had always been so, and whether any change in the relative frequency of "is/are free of" and "is/are free from" had occurred.

As the following Ngram graph shows, there has been a significant rise in frequency of the two "free of" phrases over the past 100 years, and that increase seems to have come largely at the expense of the two "free from" phrases"

Second (and even more surprising to me), I found more instances in the search results of sentences where "free of" sounded right to me and "free from" would have sounded wrong, than instances where "free from" sounded right and "free of would have sounded wrong. First and most obviously, consider the phrase "free of charge":

While here, Mr. Riddle ascertained that the transfer agencies of other western banks were conducted in some instances free of charge.

The phrase "free of charge" has always been vastly more common than "free from charge," as this Ngram graph shows:

Other instances where "free of" sounds distinctly better to my ear than "free from" include these:

It is free of the barren, sandy tracts, and great swamps, so common in the states of the south, and enjoys a richer soil and better climate than those of the north.

This date will be as soon as Lake Erie is free of ice.

Neither of these positions is free of serious objections.

When it is free of admixture with quartz and feldspar, it forms an excellent flux for iron ores.

For six inches in depth of the surface of the mounds, the soil is free of stones.

The strongest instance of "free from" I found was this:

They were indeed free from tyranny; but they wanted also that elegance which compensates for a thousand of the evils that luxury produces.

All of the preceding examples are from the nineteenth century, when "free of" was far less common than "free from" overall. In each case, the phrase "free of" means "clear of," "untainted by," or simply "without." In contrast, "free from" suggests "liberated from" or "no longer oppressed by."

The example listed above that seems to me to be least consistent with this framework is "free of ice," which usually appears in the context of geographical locations where ice is sometimes present but is absent at the time being discussed. Here, I suppose, a writer could make an argument either way: that the phrase means simply "clear of" and therefore should be "free of"; or that the phrase means "no longer fettered by" and therefore should be "free from." Both forms are still found today, though Ngram shows "free of ice" surpassing "free from ice" in frequency from 1954 forward:

Applying these subtle (and perhaps idiosyncratic) distinctions to the three examples in the original post, I would get, first,

The people were free from the barbaric dictator.

if (as the sentence implies) the dictator had once ruled them but now no longer did. I would prefer

The mashed potatoes were free from lumps.

if the mashed potatoes were originally lumpy but had subsequently had their lumps removed; but I would choose

The mashed potatoes were free of lumps.

if the mashed potatoes had come straight from the food processor lump-free, say, or if they were made from instant mashed-potato powder. And finally I would choose

I wish I could be free from this terrible cold.

because the writer is expressing a desire to escape from the cold's dominion over his body. Here, however, I like John M. Landsberg's alternative (using rid) better still.

As I said, I'm not entirely sold on this analysis, because I think most people either use "free of" and "free from" interchangeably—except in the case of "free of charge"—or arbitrarily prefer one or the other form to express the same idea, without having any finer distinctions in mind. If so, my analysis amounts to a rule in search of actual usage—a prescription rather than a description. In any event, the impressive rise of "free of" against "free from" over the past 100 years suggests that the English-speaking world has become more receptive to using "free of" in place of "free from" during that period.